A Fast, Linearized Jordan-Wigner Transform via Twisted Group Algebras

We present a unified algebraic framework relating the Heisenberg-Weyl algebra (qubits) and the Clifford algebra (fermions) through the lens of Twisted Group Algebras over $\mathbb{F}_2^{2n}$. We derive a compact, purely linear formulation of the Jordan-Wigner transform over the binary field. By introducing a specific integration operator $\mathcal{E}$, we provide explicit conversion formulas between Pauli codes and Clifford codes that include the exact sign phase via a cohomological scalar product. This formulation reduces the computational cost of the mapping from exponential matrix operations to $O(n)$ bitwise operations, making it suitable for high-performance quantum error correction simulations.